## Journal of the Brazilian Society of Mechanical Sciences

##
*Print version* ISSN 0100-7386

### J. Braz. Soc. Mech. Sci. vol.21 no.3 Rio de Janeiro Sept. 1999

#### http://dx.doi.org/10.1590/S0100-73861999000300005

**Linear-Quadratic Optimal Control of Descriptor Systems**

**Peter C. Müller **Safety Control Engineering

University of Wuppertal

D-42097 Wuppertal, Germany

mueller@wrcs1.urz.uni-wuppertal.de

Abstract

In recent years the analysis and synthesis of (mechanical) control systems in descriptor form has been established. This general description of dynamical systems is important for many applications in mechanics and mechatronics, in electrical and electronic engineering, and in chemical engineering as well. This contribution deals with linear mechanical descriptor systems and its control design with respect to a quadratic performance criterion. Here, the notion of properness plays an important role whether the standard Riccati approach can be applied as usual or not. Properness and non-properness distinguish between the cases if the descriptor system is exclusively governed by the control input or by its higher-order time-derivatives additionally. In the unusual case of non-proper systems a quite different problem of optimal control design has to be considered. Both cases will be solved completely.

Keywords.Descriptor Systems, Mechanical Systems, Linear-Quadratic Optimal Control, Causality

Introduction

In recent years the analysis and synthesis of control systems in descriptor form has been established, cf. Dai (1989). One essential class of descriptor systems are mechanical descriptor systems, i.e. finite-dimensional mechanical systems with explicit holonomic and/or nonholonomic constraints which are governed by a set of differential-algebraic equations of index 2 or 3. This special form of description of mechanical systems according to Lagrange's equation of first kind is important for many applications such as vehicle dynamics, machine dynamics, robot dynamics etc. Dealing with this type of constraint mechanical systems, stability problems have been discussed by Bajic (1992) and by Müller (1993, 1996) and first attempts of optimal control design have been performed (Müller, 1997). But some unsolved problems still exist. One of these problems is the general approach of an optimal control design of linear mechanical descriptor systems with respect to an infinite-horizon quadratic performance criterion. In this contribution, the general solution of this problem will be presented completely. Linear mechanical systems with explicit holonomic and/or nonholonomic constraints are described by

(1)

Fz = T_{2}u(2)

(3)

Here, z denotes the f-dimensional vector of displacements, u represents the r-dimensional control input, l _{1} and l _{2} are vectors of Lagrange's multipliers according to the p holonomic and q nonholonomic constraints (2) and (3). It is assumed that the f x f matrix of inertia M is symmetric and positive definite and the constraints are independent:

*rank F= p, rank G= q*(4)

In general, the constraints (2), (3) may be controlled by u, e.g. like in mechanisms. Introducing the descriptor vector

(5)

system (1-3) can be represented by

(6)

with the singular matrix

*E = diag(I _{f},I_{f}, 0,0) *(7)

and the matrices

(8)

For the control design the quadratic performance criterion

(9)

is considered where

(10)

is assumed. The problem to be solved consists in designing a (feedback) control u which minimizes (9) having regard to the dynamic system (6).

Solution of Linear Descriptor Systems

To discuss the solution of (6) the Weierstrass-Kronecker canonical representation of (6) is used (Dai, 1989). There are regular matrices R_{c}, S_{c} such that

(11)

In case of a holonomic mechanical descriptor system (1,2) these matrices are given by, cf. (Schüpphaus, 1995),

(12)

(13)

Here, L is a complementary matrix of F:

(14)

Additionally we have

(15)

Also in the case of nonholonomic mechanical descriptor systems (1,3) such transformation matrices can be determined explicitly (Schüpphaus, 1995).

Introducing

(16)

(17)

then the system representation

(18)

(19)

is derived, where N_{k} is a nilpotent matrix of order k:

(20)

The system (18) is called the "slow" subsystem and system (19) represents the "fast" subsystem. While (18) is solved in classical way, the consistent solution of (19) (cf. Dai, 1989),

(21)

shows that it generally depends on the higher-order time-derivatives of the control input, (1 £ j £ k). This is a very unusual behavior and must be regarded very carefully.

According to (21) we distinguish between proper and non-proper systems. System (6) or system (18,19) is called proper if its solution does not depend on , ..., u^{(j -1)}. The system is proper iff

*N _{k}B_{2}= 0 *(22)

Otherwise the system is non-proper. For the control design the two cases have to be considered differently. It should be mentioned that this notion of properness coincides with that of the frequency domain approach. The transfer function matrix of the descriptor system is proper if (22) holds. By the solution (21) the optimal control problem (9) can be manipulated correctly for proper and for non-proper systems.

Linear-Quadratic Optimal Control Design

Firstly, we have to rewrite the performance criterion (9,10) regarding the coordinate transformation (16) into the Weierstrass-Kronecker canonical representation (18,19). Then the criterion (9) becomes

(23)

where

(24)

The optimization of (23) has to be performed under the conditions (18) and (21). According to the properness or non-properness of the descriptor system, two different optimization problems have to be considered.

Proper descriptor systems

If (22) holds the solution

*x _{2} = -B_{2}u*(25)

is used to replace the x_{2} variables in the performance criterion (23):

(26)

For proper descriptor systems a standard linear-quadratic optimal control problem has been derived. The control of the slow subsystem (18) has to be designed with respect to the criterion (26). Therefore, the solution is obtained by the "Riccati" procedure for a system of reduced order n1=dim (x1). The optimal proportional feedback control is

(27)

where P_{1} is the unique positive (semi-)definite solution of the algebraic Riccati equation

(28)

where the matrices

(29)

(30)

(31)

have been used.

For proper descriptor systems a standard solution of a reduced order system has been established.

Non-proper descriptor systems

For non-proper problems an extension of state and control variables has to be carried out:

(32)

*v = u ^{(j-l)}*(33)

Then (21) is replaced by

(34)

Here, v is considered as a new control input vector. Introducing an extended state vector

(35)

an extended dynamical system can be described including the dynamics of the slow subsystem and of the extensions (32,33):

(36)

(37)

It is not dificult to show that the controllability of system (36,37) is guaranteed if the slow subsystem (18) is controllable.

The solution (21) of the fast subsystem (19) can be written as

(38)

with

(39)

Substituting (38) in the performance criterion (23), a linear-quadratic optimal control problem with respect to x_{e} and v appears. But obviously a singular control problem is obtained. There is not a regular weighting of the new control input v. To regularize the problem it is recommended to introduce such a weighting (R_{v}). But then the question may arise whether it is reasonable to introduce additional weightings (Qx ) of the variables x _{e} . By this, finally a modified performance criterion is defined:

(40)

where the weighting matrices are given by

(41)

(42)

. (43)

The optimization problem for the non-proper descriptor system is defined by (36) and (40). Therefore, now we have a standard linear-quadratic optimal control problem for the extended system (36) of order

*n _{e} = n_{1} + r(j-l) *(44)

The solution is represented by

(45)

where P_{e} is the unique positive (semi-)definite solution of the algebraic Riccati equation

(46)

where

(47)

(48)

Under very weak conditions of (A_{1}; B_{1})-stabilizability and ()-detectability the resulting closed-loop control system

(49)

with the system matrix

(50)

is asymptotically stable.

Now, we have in mind the original meaning of the extended variables. Defining the gain matrix of (45) as

(51)

then the result (45) represents a dynamic feedback control

(52)

for the slow subsystem (18),

According to the asymptotic stability of (49) the coupled system (18,52) is asymptotically stable. The over all dimension of the optimally controlled system is ne (44).

Illustrative Examples

In the following three small academic examples are discussed to illustrate the question of proper-ness and optimization.

Proper mechanical descriptor system

The first example consists of two spring-mass-oscillators where the masses are connected by a rigid bar. Additionally the first oscillator is controlled. Lagrange's equations of first kind are the following:

(53)

(54)

*z _{1} - z_{2} = 0*(55)

The description (5-8) is obtained by

(56)

. (57)

It is easily chequed that the system is proper. There is a more general result behind. Mechanical descriptor systems (1,2,3) are proper iff T_{2} = 0, T_{3} = 0, i.e. if the constraints are not controlled. If at least one constraint is controlled, i.e. it depends explicitly on u, then the system (1,2,3) is non-proper.

The optimal control design is performed corresponding to Proper descriptor systems. In (Müller, 1997) the linear-quadratic optimal regulator problem has been discussed. There, well-known results have been obtained, but this time based on a DAE-approach and not on the basis of a classical state-space approach which arises from the corresponding ordinary differential equation

(58)

where z1 has been chosen as a generalized coordinate

Non-proper mechanical descriptor system

The second example differs from the first one by replacing the control to the constraint such that we have the typical problem of a controlled mechanism:

(59)

(60)

*z _{1} - z_{2} * +

*u = 0*(61)

The description (5-8) is obtained by (56) and A as in (57) but a different B:

*B ^{T} = [0 0 0 0 1].* (62)

The condition (22) is violated; therefore this system is non-proper. This fact can be shown explicitly if we write down Lagrange's equation of second kind with respect to the generalized coordinate z1 and the equation determining the constraint force l :

(63)

(64)

The representation (63,64) shows that the introduction of extended state variables (32,35) and of a new control input v = ü is necessary to handle properly the optimization procedure.

Kinematic control of a mass

A one-dimensional motion of a single mass will be kinematically controlled. The motion is governed by the descriptor system

(65)

*z(t) = u(t)* (66)

For this very simple example usually the algebraic equation (66) will be considered alone. But here the dynamic efects are included by the differential equation (65). Defining x_{1} = l /m, x_{2} = , x_{3} = z the description of a fast subsystem (19) is obtained:

(67)

with

(68)

The solution is verified according to (21) as

(69)

The control should be designed with respect to

(70)

Regarding the discussion of section 3.2 the additional variables are introduced:

(71)

This leads to the notation

(72)

of the criterion (70) which leads to a singular control problem. The regularization is carried out by adding an additional weighting term with respect to v:

(73)

Now we have a standard optimal control problem minimizing the criterion (73) for the dynamic system (71). The related Riccati equation (46) has the solution

(74)

The closed-loop control system is governed according to (52) by

(75)

showing asymptotically stable behavior.

Although this is a very simple example, it demonstrates the necessity of the extended variables and of regularized performance criterion.

References

Bajic, V. B.,1992, "Lyapunov's Direct Method in the Analysis of Singular Systems and Networks", Shades Technical Publications, Hillcrest, Natal, RSA. [ Links ]

Dai, L.,1989, "Singular Control Systems", Lecture Notes in Control and Information Sciences Vol. 118, Springer,Berlin-Heidelberg. [ Links ]

Müller, P. C.,1993, "Stability of Linear Mechanical Systems with Holonomic Constraints", Appl. Mech. Rev. 46, no. 11, part 2, pp. S160-S164. [ Links ]

Müller, P. C.,1996, "Stability of Nonlinear Descriptor Systems", Z. Angew. Math. Mech. 76, Supplement 4, pp. 9-12. [ Links ]

Müller, P. C.,1997, "Optimal Control of Mechanical Descriptor Systems", In: D.van Campen (ed.), Interaction Between Dynamics and Control in Advanced Mechanical Systems, Kluwer Academic Publisher, Dordrecht, to appear. [ Links ]

Schüpphaus, R.,1995, "Regelungstechnische Analyse und Synthese von Mehrk· orpersystemen in Deskriptorform" (in German), VDI-Fortschr.-Ber., Reihe 8, Nr. 478, VDI-Publisher, Düsseldorf. [ Links ]

Presented at DINAME 97 - 7th International Conference on Dynamic Problems in Mechanics, 3 - 7 March 1997, Angra dos Reis, RJ, Brazil. Technical Editor: Agenor de Toledo Fleury.